
die Beteiligung
participation
Active participation or involvement in an activity, event, or process.
Die Beteiligung der Bürger an dem Projekt war sehr hoch.
Citizen participation in the project was very high.
participation / stake
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participation
Active participation or involvement in an activity, event, or process.
Die Beteiligung der Bürger an dem Projekt war sehr hoch.
Citizen participation in the project was very high.

stake
A financial share that a person or company owns in another company.
Der Investor hat seine Beteiligung an dem Start-up auf 25 Prozent erhöht.
The investor increased his stake in the start-up to 25 percent.
'Die Beteiligung' (participation/stake) is derived from the verb 'sich beteiligen' (to participate), which in turn goes back to the noun 'der Teil' (part). It describes the action of becoming a 'Teil' (part) of something or having a 'Teil' (share) in something.
Alle können sich an der Diskussion beteiligen.
Everyone can participate in the discussion.
To say what someone participates in or what someone owns a share in, the preposition 'an' with the dative is almost always used. This structure applies to both main meanings of the word.
Die Beteiligung an der Wahl war dieses Jahr sehr niedrig.
Participation in the election was very low this year.
You can tell from the context whether participation or a financial stake is meant. If it is about projects, elections, or discussions, participation is meant. If it is about companies, investors, and percentages, a financial stake is meant.
Wir freuen uns über die rege Beteiligung an unserem Sommerfest.
We are pleased about the lively participation in our summer festival.
When it is about participation, 'Beteiligung' is often combined with adjectives such as 'aktiv' (active), 'rege' (lively), 'hoch' (high), or 'gering' (low) to describe the extent of involvement.
Eine aktive Beteiligung der Eltern ist in der Schule sehr erwünscht.
Active participation by parents is very welcome at the school.
share / proportion
When referring to ownership in a company, both words can fit. Anteil mainly names the specific size or individual part of the ownership; Beteiligung refers more to the financial commitment or ownership relationship as a whole. Anteil is therefore usually more appropriate for a precise percentage.

Ihre Beteiligung an dem Unternehmen besteht seit fünf Jahren.
She has had a stake in the company for five years.
Ihr Anteil an dem Unternehmen beträgt zwanzig Prozent.
Her share in the company is twenty percent.